Measurement techniques for ultra-short optical pulses in the picosecond and femtosecond range typically involve all-optical methods, most commonly based on the second harmonic generation (SHG) optical nonlinearity. For many applications, such as measurement of low power signals in lightwave communications, it is desirable to reduce the power required to characterize the pulse. Simultaneously, the measurement must provide sufficient optical bandwidth to avoid measurement distortion. Usually optical bandwidth is increased by decreasing the length of the nonlinear crystal responsible for the SHG. However, this reduces the efficiency of the nonlinear optical interaction and leads to increased power requirements. The result is that there is a trade-off between optical bandwidth and measurement sensitivity; increasing the optical bandwidth to avoid measurement distortion leads to an undesirable decrease in measurement sensitivity.
Phase matching between the fundamental and second harmonic signals is required in order to obtain the highest efficiency for SHG. However, due to the phase velocity difference between the input wave and the generated second harmonic wave, phase matching does not usually occur naturally. Microstructuring of the medium may be employed to achieve phase matching between the input signal and the harmonic wave.
For continuous-wave (narrow bandwidth) input signals, the microstructuring can take the form of a periodic patterning of the nonlinear optical susceptibility. A material that may be fabricated to employ such periodic patterning of the nonlinear susceptibility is periodically poled lithium niobate (PPLN), in which the orientation direction of crystal domains is periodically modulated, typically along the interaction direction, during the fabrication process. The optical bandwidth for second harmonic generation, formally known as the phase matching bandwidth, is inversely proportional to the length of the nonlinear crystal, both in the case of a uniform crystal and in the case of a quasi-phase-matched crystal with periodic microstructuring.
Several studies investigating quasi-phase-matching (QPM) via nonperiodic microstructuring of the nonlinear medium have also been reported including lithium niobate crystals where the poling period varies along the length of the crystal. Such crystals are referred to as aperiodically poled lithium niobate (A-PPLN). The modulation of the quasi-phase-matching or poling period broadens the phase matching bandwidth for SHG and the optical bandwidth can be chosen largely independently of the crystal length, which is not the case with uniform or periodically poled nonlinear crystals.
In addition, the efficiency of the SHG process can be increased by increasing the nonlinear crystal length. For the case of a continuous-wave (narrowband) laser tuned for perfect phase matching, the efficiency can increase with the square of the crystal length. For sufficiently short pulse (broadband) lasers, the efficiency increases in proportion to the crystal length.
To obtain accurate results in autocorrelation, frequency resolved optical gating (FROG), and other ultrashort pulse measurement techniques based on SHG, the phase matching bandwidth for SHG should exceed the optical bandwidth of the signal of interest. This condition is usually met by reducing the length of the nonlinear crystal. If the crystal length is reduced by N, the phase matching bandwidth is increased by N; however the peak efficiency drops by N2, which means that there is a large cost in sensitivity.
In second-order nonlinear optics, e.g., second harmonic generation (SHG), material dispersion causes the phase between the input signal electric field and the electric field at the newly generated frequency to drift with distance along the crystal, preventing continuous growth of the newly generated field. The distance over which the accumulated phase difference between the second harmonic and the driving polarizations changes by π is called the coherence length lc. In QPM, continuous growth of the generated field along the propagation direction is achieved by resetting the phase of the driving polarization every coherence length by changing the sign of the nonlinear coefficient χ. In the Fourier domain (wave-vector space), QPM is equivalent to compensating the wave-vector difference between the nonlinear polarization and the second harmonic field wave by applying a Fourier component of a grating with appropriate period Λg=2lc.
In ferroelectric materials, such as lithium niobate, the sign of the second order nonlinearity is related to the crystal orientation; alternation of the sign of the nonlinearity, and hence QPM, is achieved by periodic poling. A method of periodic poling consists of applying a periodic electric field pattern on the ferroelectric wafer through a dielectric mask causing reversal of the domain orientation under the surface of the electrodes. The periodic crystal orientation remains permanently after removal of the poling field. A dielectric mask may be prepared lithographically which leads to high resolution as well as precise positioning.